Skip to content
BY 4.0 license Open Access Published by De Gruyter March 31, 2023

Existence and Finite-Time Stability Results for Impulsive Caputo-Type Fractional Stochastic Differential Equations with Time Delays

  • Mengquan Tian and Danfeng Luo EMAIL logo
From the journal Mathematica Slovaca

ABSTRACT

This paper mainly discusses the existence and finite-time stability of solutions for impulsive fractional stochastic differential equations (IFSDEs). By applying the Picard-Lindelöf iteration method of successive approximation scheme, we establish the existence results of solutions. Subsequently, the uniqueness of solution is derived by the method of contradiction. In addition, we investigate the finite-time stability by means of the generalized Grönwall-Bellman inequality. As an application, examples are provided to expound our theoretical conclusions.

2020 Mathematics Subject Classification: 26A33; 65C30; 34K20

1. Introduction

Due to its hereditary properties, fractional calculus has been studied extensively in recent decades by researchers, such as fractional physics, chaos and turbulence, viscoelasticity and non-newtonian fluid mechanics, decoupling of polymer materials, automatic control theory, chemical physics, random process, diffusion, etc. For more knowledge in details, one can refer to relevant monographs [35, 38].

As is known to us, the deterministic system is often unstable due to noise, and a natural idea is adding random parameters to it. Unstable deterministic system may become stable system due to the involvement of random terms. Then this kind of equation provides a more accurate description of model phenomena and the theory of fractional stochastic differential equations (FSDEs) has been extensively studied and it has developed quickly. It mainly arises from environmental changes, ecological, financial markets and many other areas. There are series of literatures that have documented this kind of system, such as [34, 39].

In addition, impulses effects also play an indispensable role in practical application, which means that the state of equations is subjected to abrupt and short changes. Impulsive differential equations can be describing some natural phenomena, such as biology, harvesting, social microsystems, diseases, and so on. For more theory about impulsive system and its detail, one can refer to [5, 6, 12, 13, 16, 22, 43]. IFSDEs have also attracted increasing interests amongst researchers due to their more practical applications, such as the research of telecommunications, finance, electronics, economics, and mechanics. In the actual situation we can hardly neglect the existence of impulses effects. Therefore, the effects of impulses have been taken into account by many researchers in their study of FSDEs [1, 7, 14, 18].

A primary research area in practical applications of FSDEs is the study of existence and uniqueness of the equations. A large amount of literatures has proposed various methods to solve this kind of problem, such as [37, 48], and so on. Recently, Wang et al. [48] deduced the existence and uniqueness results of FSDEs through the method of Carathéodory approximation, which assumes that the coefficients satisfy non-Lipschitz condition. Moghaddam et al. [37] derived the existence and uniqueness results of the FSDEs by virtue of the classical Picard iteration method and the Grönwall inequality, supposing that the coefficients satisfy global Lipschitz condition. In [42], Umamaheswari et al. considered the fractional stochastic differential system with Lévy noise; this system is more generalized than the previous two, in which the results are obtained by classical Picard-Lindelöf method and assume the coefficients satisfying Lipschitz condition. One can refer to Ahmadova et al. [4] and Zada et al. [51] for more details.

By adding impulses item to fractional differential equations, we will encounter what kinds of difficulties are in dealing with equations. It is worth to mention the related work about impulsive fractional differential equations. In [15], Guo et al. investigated the impulsive fractional functional differential equations, in which the existence and uniqueness results were deduced by applying Schauder fixed point theorem and Banach contraction principle methods, respectively. Abouagwa et al. [1] researched impulsive FSDEs, in which the random process is an independent fractional Brownian motion; the main results were deduced by means of Carathéodory type approximation. In references [4446], Wang et al. researched several kinds of impulsive fractional Cauchy problem. Zada et al. [3] researched the existence and uniqueness of solutions for nonlinear stochastic weighted impulsive ψ-Hilfer neutral integro-fractional differential delay system with the aid of fixed point theorems and Banach contraction principle.

Finite-time stability analysis is also another dominant topic in the study of fractional differential equations, because the stability analysis of models is essential in practical applications. Unlike Lyapunov stability analysis, finite-time stability is considered in a system, in which an interval is finite. The study of finite-time stability can be divided into two kinds: one is to study the convergence of system to an equilibrium state in finite time and estimate its resting time in the category of traditional stability; the other is to research the transient performance of system in finite time domain, independent of the general stability problem. In this paper we consider the latter one. The research of stability of fractional differential equations has become a hot subject and attracts increasing interests from many scholars and engineers, such as [17], that researched finite-time stability of impulsive fractional difference equations, in which delay terms and the theoretical results are derived by the generalized Grönwall inequality. In [29], the authors researched a class of ψ-Hilfer fractional differential equations, in which delay terms and impulses effects were considered. Uniqueness and existence results were derived by Banach contraction principle methods and Schauder’s fixed point theorem, respectively. In [53], a new impulsive type of Grönwall inequality was applied to deduce the finite-time stability of the considered model. In [32, 33, 47, 50, 52], the authors obtained finite-time stability results by means of delayed exponential matrix method. Fore more details one can refer to papers [811,20,21,24,28,30,40].

In this article, due to the lack of results about the existence of solution for IFSDEs, inspired by [15, 17, 37, 42, 48] and other mentioned papers, the following impulsive Caputo-type FSDEs with time delays is considered:

1.1 {CD0+qY(t)=ξ(t,Y(t),Y(tς))+ζ(t,Y(t),Y(tς))dw(t)dt,t[0,T], ttk,Y(tk+)=Y(tk)+Ik(Y(tk)),k=1,2,,m,Y(t)=Φ(t),t[ς,0],

where CD0+q is the Caputo fractional derivative of order q(12,1), ξ:[0,T]×Rp×RpRp, ζ:[0,T]×Rp×RpRp×d are continuous functions. ℝp denotes the Euclidean space with norm |·|, w(t) denotes standard Brownian motion with d-dimensional. The positive constant ς ∈ ℝ+ represents the delay. Here, Ik:RpRp are continuous functions with fixed times tk satisfying 0=t0<t1<t2<<tm<tm+1=T, Y(t) is left continuous at t = tk, Y(tk+) and Y(tk) are the right and left limits of Y(t), respectively. 𝔼 represents mathematical expectation, Φ(t)C([ς,0];Rp) satisfying E(|Φ0|2)<, where Φ0 = Φ(0).

Compared with relate references mentioned in the literatures, such as [2, 23, 27, 37, 42, 48] and others, the major contributions of this article contain the following three aspects:

  1. In [27] finite-time stability results were deduced by applying Lyapunov function approach, and in [23, 25, 26] stability results were deduced by applying delayed Mittag-Leffler type matrix. In this article, finite-time stability of FSDEs with constant time delays is obtained by applying the generalized Grönwall-Bellman inequality.

  2. Compared with [23, 37, 48], we consider a more general model, in which impulsive effects are added. Moreover, the generalized model is more practical.

  3. In [2, 48], the existence and uniqueness results were deduced by applying the Carathéodory approximation. In this paper, a novel method based on classical Picard-Lindelöf method and the generalized Grönwall-Bellman inequality is exploited to derive the existence and uniqueness results for the impulsive Caputo-type FSDEs with time delays. Those results will enrich the relevant literatures in dealing with IFSDEs.

The vein of this article is developed as follows: in Section 2, we will give a few well-know concepts and introduce some lemmas, which are useful to our work. Moreover, we give the equivalent fractional integral equations of solutions for the system (1.1). Section 3 is aimed to derive the existence of solutions for impulsive Caputo-type FSDEs with time delays. In Section 4, finite-time stability results of the addressed equations are researched. In order to expound the usefulness of proposed theoretical results, we show two examples in Section 5.

2. Preliminaries

In this part, we will present a few well-known concepts and introduce some lemmas, which play an indispensable role in our derivation.

Definition 1

([2]). Riemann-Liouville integral of function ϕ(t) with order q > 0 can be written as

I0+qφ(t)=1Γ(q)0t(tτ)q1φ(τ)dτ,

where t > 0 and Γ(·) is the Eulers Gamma function.

Definition 2

([19]). The Caputo derivative of order q > 0 can be written as

CD0+qψ(t)=1Γ(mα)0t(tμ)mq1ψ(m)(μ)dμ,

where m − 1 < q < m, m ∈ ℕ. Particularly, for q ∈ (0, 1),

I0+qCD0+qψ(t)=ψ(t)ψ(0).

Definition 3

([15]). The Riemann-Liouville derivative operator of order q > 0 can be denoted as

LDtqϕ(t)=1Γ(mq)dmdtm0tϕ(ν)(tν)q+1mdν,m1<q<m.

Definition 4

([15]). The Caputo derivative of order q > 0 can be denoted as

cDtqϕ(t)=LDq[ϕ(t)k=0m1tkk!ϕ(k)(0)],m1<q<m.

Remark 1.

  1. suppose that f(t)Cm[0,), then

    cDtqf(t)=1Γ(mq)0tf(m)(s)(ts)q+1mds=Itmqf(m)(t),t>0, m1<q<m.
  2. cDtqC0, where C is a constant.

Lemma 2.1.

Let ϱ ∈ (0, 1) and ϕ(t) be continuous, η is a constant. A function Y(·) is a solution of the following equation

Y(t)=η1Γ(ϱ)0a^(a^v)ϱ1φ(v)dv+1Γ(ϱ)0t(tv)ϱ1φ(v)dv

if and only if Y(·) is a solution of the following Cauchy problem

{cDtqY(t)=φ(t),t[0,T],Y(a^)=η,a^(0,T).

Proof.

Taking Caputo derivative on both sides of Y(t), from Remark 1 we get

cDtqY(t)=φ(t).

Letting t = â, we have

Y(a^)=η1Γ(ϱ)0a^(a^v)ϱ1φ(v)dv+1Γ(ϱ)0a^(a^v)ϱ1φ(v)dv=η.

On the other side,

I0+qCD0+qY(t)=I0+qφ(t),

then

Y(t)=C+1Γ(ϱ)0t(tv)ϱ1φ(v)dv,

where C represent a constant. Thus

Y(a^)=C+1Γ(ϱ)0a^(a^v)ϱ1φ(v)dv=η,

then we obtain

C=η1Γ(ϱ)0a^(a^v)ϱ1φ(v)dv.

Therefore

Y(t)=η1Γ(ϱ)0a^(a^v)ϱ1φ(v)dv+1Γ(ϱ)0t(tv)ϱ1φ(v)dv.

This completes the proof. □

Lemma 2.2.

A function Y(·) is a solution of (1.1) if and only if Y(·) is a solution of the following fractional integral equations

2.1 Y(t)={ Φ(t),t[ς,0],Φ0+1Γ(q)0t(ts)q1ξ(s,Y(s),Y(sς))ds+1Γ(q)0t(ts)q1ζ(s,Y(s),Y(sς))dw(s),  t(0,t1],Φ0+1Γ(q)0t(ts)q1ξ(s,Y(s),Y(sς))ds+1Γ(q)0t(ts)q1ζ(s,Y(s),Y(sς))dw(s)+k=1nIk(Y(tk)),t(tn,tn+1],  n=1,2,m.

Proof.

The proof is similar to the proof of Lemma 2.7 in [15]. □

Definition 5

(Chebyshev’s inequality, [42]). For a random variable θ and 1 ≤ q < ∞, the following inequality holds

P(|θ|γ)1γqE(|θ|q) for all γ>0.

Definition 6

([29]). Given positive numbers δ, σ satisfying δ < σ, then system is finite-time stable if the following hold

E(supςt0|Φ(t)|2)δE(sup0tT|Y(t)|2)σfor all t[0,T].

Lemma 2.3.

(Doob’s martingale inequalities, [37]). Let [a, b] is a bounded interval within+ and {Yt}t ≥ 0 bed-valued martingale. Assume that p > 1 and {Yt}t0LP(Ω;Rd), then

E(supt[a,b]|Yt|p)(pp1)pE(|Yb|p).

Lemma 2.4.

(Borel Cantelli lemma, [42]). If {Ak} ⊂ ℱ and k=1P(Ak)< , then

P(limksupAk)=0.

Lemma 2.5.

(Jensen’s inequality, [31]). Let m ∈ ℕ and x1, x2, …, xm be nonnegative real numbers, then

(j=1mxj)pmp1j=1mxjp for p>1.

Lemma 2.6.

([36, 41]). Let ϑ(t) is a piecewise continuous and nonnegative function, satisfying the following

ϑ(t)C+t0tV(τ)ϑ(τ)dτ+t0<τi<tβiϑ(τi) for tt0,

where C ≥ 0, βi ≥ 0, V(τ) > 0, and τi represent the discontinuity points of the function ϑ(t). Then the following inequality holds

ϑ(t)Ct0<τi<t(1+βi)exp[t0tV(τ)dτ].

Lemma 2.7.

(Generalized Grönwall-Bellman inequality, [49]). Let 0 < q < 1 and time interval I := [0, T), where T ≤ ∞. A locally integrable and nonnegative functions α(t), β(t), and γ(t) are nondecreasing, nonnegative bounded continuous functions on I. If ϑ(t) is locally integrable on I, nonnegative and satisfies the following

ϑ(t)α(t)+β(t)0tϑ(s)ds+γ(t)0t(ts)q1ϑ(s)ds,

then the following estimation holds

ϑ(t)α(t)+n=1i=0n(ni)βni(t)[γ(t)Γ(q)]iΓ(iq+ni)0t(ts){i(2q1)(i+1n)}α(s)ds.

3. Existence results

In this part, we are going to prove the existence and uniqueness results of the system (1.1). Here, the results are obtained by using Picard-Lindelöf method of successive approximation scheme and impulsive Grönwall inequality. Before showing the theoretical results, we first give a couple of hypotheses:

  1. Assume that ξ and ζ satisfy the following conditions, for any θ1, θ2, ϑ1, ϑ2 ∈ ℝp and for all t ∈ [0, T],

    |ξ(t,θ1,ϑ1)ξ(t,θ2,ϑ2)|2|ζ(t,θ1,ϑ1)ζ(t,θ2,ϑ2)|2L1(|θ1θ2|2+|ϑ1ϑ2|2)

    and

    |ξ(t,θ,ϑ)|2ζ(t,θ,ϑ))|2L2(1+|θ|2+|ϑ|2),

    where L1 and L2 are real and positive constants.

  2. There exist some constants dkR+(k=1,2,) satisfying

    |Ik(x)Ik(y)|dk|xy|

    for all x, y ∈ ℝp and |Ik(0)| = 0.

Theorem 3.1.

Suppose that hypotheses (1) and (2) hold, then system (1.1) has a unique solution on [−ς, T].

Proof.

Existence: Assuming that T > 0 is sufficiently small, then T > 0 satisfying

3.1 ε=[2(3T+12)L1T2q1(2q1)Γ2(q)+3mk=1mdk2]<12.

Define Y(0)(t) = Y0 = Φ0, Y(l)(t) = Φ(t) for all t ∈ [−ς, 0], l = 0, 1, 2,…. For all t ∈ (tn, tn + 1], 0 ≤ nm, let us define inductively as follows:

3.2 Y(l+1)(t)={ Φ0+1Γ(q)0t(ts)q1ξ(s,Y(l)(s),Y(l)(sς))ds+1Γ(q)0t(ts)q1ζ(s,Y(l)(s),Y(l)(sς))dw(s),t(0,t1],Φ0+1Γ(q)0t(ts)q1ξ(s,Y(l)(s),Y(l)(sς))ds+1Γ(q)0t(ts)q1ζ(s,Y(l)(s),Y(l)(sς))dw(s)+k=1nIk(Y(l)(tk)),t(tn,tn+1],n=1,2,m.

For all t ∈ (tn, tn + 1], 0 ≤ nm, we obtain by virtue of Jensen’s inequality, Cauchy-Schwarz inequality, Itô isometry and hypotheses (1)–(2),

E(|Y(l+1)(t)|2)4E(|Φ0|2)+4TΓ2(q)E(0t(ts)2q2|ξ(s,Y(l)(s),Y(l)(sς))|2ds)+4Γ2(q)E(0t(ts)2q2|ζ(s,Y(l)(s),Y(l)(sς))|2ds)+4E(|k=1nIk(Y(l)(tk))|2)4TL2T2q1(2q1)Γ2(q)+4TL2Γ2(q)0t(ts)2q2E(|Y(l)(s)|2+|Y(l)(sς)|2)ds+4L2T2q1(2q1)Γ2(q)+4L2Γ2(q)0t(ts)2q2E(|Y(l)(s)|2+|Y(l)(sς)|2)ds+4E(|Φ0|2)+4nk=1ndk2E(|Y(l)(tk)|2)4E(|Φ0|2)+4(T+1)L2T2q1(2q1)Γ2(q)+4(T+1)L2T2q1(2q1)Γ2(q)Φ+8(T+1)L2Γ2(q)0t(ts)2q2E(|Y(l)(s)|2)ds+4mk=1mdk2E(|Y(l)(tk)|2).

Therefore, for an arbitrary integer h ≥ 1, we have

maxl[0,h]E(|Y(l+1)(t)|2)C1+8(T+1)L2Γ2(q)0t(ts)2q2maxl[0,h]E(|Y(l)(s)|2)ds+4mk=1mdk2maxl[0,h]E(|Y(l)(tk)|2),

where

C1=4E(|Φ0|2)+4(T+1)L2T2q1(2q1)Γ2(q)+4(T+1)L2T2q1(2q1)Γ2(q)Φ.

Note that

maxl[0,h]E(|Y(l)(s)|2)=max{E(|Φ0|2),E(|Y(1)(s)|2),,E(|Y(h)(s)|2)}max{E(|Φ0|2),maxl[0,h]E(|Y(l+1)(s)|2)}(E|Φ0|2)+maxl[0,h]E(|Y(l+1)(s)|2).

Hence

maxl[0,h]E(|Y(l+1)(t)|2)C1+8(T+1)L2Γ2(q)0t(ts)2q2(E(|Φ0|2)+maxl[0,h]E(|Y(l+1)(s)|2))ds+4mk=1mdk2(E(|Φ0|2)+maxl[0,h]E(|Y(l+1)(tk)|2))C1+8(T+1)L2T2q1(2q1)Γ2(q)E(|Φ0|2)+4mk=1mdk2E(|Φ0|2)+8(T+1)L2Γ2(q)0t(ts)2q2maxl[0,h]E(|Y(l+1)(s)|2)ds+4mk=1mdk2maxl[0,h]E(|Y(l+1)(tk)|2).

Due to the Lemma 2.6, we obtain

maxl[0,h]E(|Y(l+1)(t)|2)C2k=1m(1+4mdk2)exp[8(T+1)L2T2q1(2q1)Γ2(q)],

where C2=C1+8(T+1)L2T2q1(2q1)Γ2(q)E(|Φ0|2)+4mk=1mdk2E(|Φ0|2), then we have

E(|Y(l+1)(t)|2)C2k=1m(1+4mdk2)exp[8(T+1)L2T2q1(2q1)Γ2(q)],

for any 0 ≤ l and t ∈ (tn, tn + 1], 0 ≤ nm.

Next, from (3.2) we derive

Y(l+1)(t)Y(l)(t)=1Γ(q)0t(ts)q1[ξ(s,Y(l)(s),Y(l)(sς))ξ(s,Y(l1)(s),Y(l1)(sς))]ds+1Γ(q)0t(ts)q1[ζ(s,Y(l)(s),Y(l)(sς))ζ(s,Y(l1)(s),Y(l1)(sς))]dw(s)+k=1n[Ik(Y(l)(tk))Ik(Y(l1)(tk))].

Therefore, by applying Jensen’s inequality, Cauchy-Schwarz inequality, Doob’s martingale inequality and hypotheses (1)–(2), we obtain

E(sup0ut|Y(l+1)(u)Y(l)(u)|2)3Γ2(q)E(sup0ut|0u(us)q1[ξ(s,Y(l)(s),Y(l)(sς))ξ(s,Y(l1)(s),Y(l1)(sς))]ds|2)+3Γ2(q)E(sup0ut|0u(us)q1[ζ(s,Y(l)(s),Y(l)(sς))ζ(s,Y(l1)(s),Y(l1)(sς))]dw(s)|2)+3E(sup0ut|k=1n[Ik(Y(l)(tk))Ik(Y(l1)(tk))]|2)3TΓ2(q)0t(ts)2q2E(|ξ(s,Y(l)(s),Y(l)(sς))ξ(s,Y(l1)(s),Y(l1)(sς))|2)ds+12Γ2(q)0t(ts)2q2E(|ζ(s,Y(l)(s),Y(l)(sς))ζ(s,Y(l1)(s),Y(l1)(sς))|2)ds+3nk=1ndk2E(sup0ut|Y(l)(uk)Y(l1)(uk)|2),

then we get

E(sup0ut|Y(l+1)(u)Y(l)(u)|2)(3T+12)L1Γ2(q)0t(ts)2q2E(|Y(l)(s)Y(l1)(s)|2+|Y(l)(sς)Y(l1)(sς)|2)ds+3nk=1ndk2E(sup0ut|Y(l)(uk)Y(l1)(uk)|2)2(3T+12)L1Γ2(q)0t(ts)2q2E(sup0ηs|Y(l)(η)Y(l1)(η)|2)ds+3nk=1ndk2E(sup0ut|Y(l)(u)Y(l1)(u)|2)2(3T+12)L1T2q1(2q1)Γ2(q)E(sup0ut|Y(l)(u)Y(l1)(u)|2)+3nk=1ndk2E(sup0ut|Y(l)(u)Y(l1)(u)|2)[2(3T+12)L1T2q1(2q1)Γ2(q)+3mk=1mdk2]E(sup0ut|Y(l)(u)Y(l1)(u)|2).

Now, for l = 0,

E(sup0ut|Y(1)(u)Y(0)(u)|2)3Γ2(q)E(sup0ut|0u(us)q1ξ(s,Y(0)(s),Y(0)(sς))ds|2)+3Γ2(q)E(sup0ut|0u(us)q1ζ(s,Y(0)(s),Y(0)(sς))ds|2)+3E(|k=1nIk(Y(0)(tk))|2)3TΓ2(q)E(0t(ts)2q2|ξ(s,Y(0)(s),Y(0)(sς))|2ds)+12Γ2(q)E(0t(ts)2q2|ζ(s,Y(0)(s),Y(0)(sς))|2ds)+3nk=1ndk2E(|Y(0)(tk)|2).

Due to the hypothesis (1), we get

E(sup0ut|Y(1)(u)Y(0)(u)|2)3TL2Γ2(q)E(0t(ts)2q2(1+|Y(0)(s)|2+|Y(0)(sς)|2)ds)+12L2Γ2(q)E(0t(ts)2q2(1+|Y(0)(s)|2+|Y(0)(sς)|2)ds)+3nk=1ndk2E(|Φ|2)(3TL2+12L2)T2q1(2q1)Γ2(q)+[2(3TL2+12L2)T2q1(2q1)Γ2(q)+3mk=1mdk2]E(|Φ|2).

Thus, by virtue of mathematical induction, we derive

E(sup0ut|Y(l+1)(u)Y(l)(u)|2)Cεl,l=0,1,,t(tn,tn+1], 0nm,

where

C=E(sup0ut|Y(1)(u)Y(0)(u)|2),ε=[2(3T+12)L1T2q1(2q1)Γ2(q)+3mk=1mdk2],

which are constants depending only on q, T, L1, L2, dk, m. Then we can get from Chebyshev’s inequality

P{supu[0,t]|Y(l+1)(u)Y(l)(u)|212l}C(2ε)l.

Summing up the resultant inequalities, we obtain

l=0P{supu[0,t]|Y(l+1)(u)Y(l)(u)|212l}l=0C(2ε)l.

By means of Borel Cantelli lemma and Weierstrass convergence criterion, we know that l=0C(2ε)l<. We conclude that supu[0,t]|Y(l+1)(u)Y(l)(u)|2 converges to 0, which implies that the successive approximation {Y(l)} is a Cauchy sequence. Therefore Y(l) converges, almost surely, uniform on [−ς, T] to a limit Y(t) defined by

limL(Y(0)(t)+l=1L(Y(l)(t)Y(l1)(t)))=limLY(L)=Y(t).

Then Y(t) is continuous. Using Fatou’s lemma and the bound of E(|Y(l+1)(t)|2), we get

E(|Y(t)|2)C2k=1m(1+4mdk2)exp[8(T+1)L2T2q1(2q1)Γ2(q)].

From (3.2), we derive

3.3 Y(t)={Φ(t),t[ς,0],Φ0+1Γ(q)0t(ts)q1ξ(s,Y(s),Y(sς))ds+1Γ(q)0t(ts)q1ζ(s,Y(s),Y(sς))dw(s),t(0,t1],Φ0+1Γ(q)0t(ts)q1ξ(s,Y(s),Y(sς))ds+1Γ(q)0t(ts)q1ζ(s,Y(s),Y(sς))dw(s)+k=1nIk(Y(tk)),t(tn,tn+1],n=1,2,,m.

In order to eliminate the limitation posed by (3.1), letting γ > 0 sufficiently small, satisfying

3.4 ε=[2(3γL2+12L1)γ2q1(2q1)Γ2(q)+3mk=1mdk2]<12.

Then the considered system has at least one solution on [−ς, γ]. Considering on interval [γ, 2γ] and repeating the process as above, it is easy to obtain that for the system also exists a solution on [−ς, T]. The existence of the solutions is proved.

Uniqueness: Let Y(t), U(t) represent two different solutions of system (1.1), consider that Y(t) = U(t) = Φ(t) for all t ∈ [−τ, 0]. It is easy to derive for all t ∈ [0, T],

Y(t)U(t)=1Γ(q)0t(ts)q1[ξ(s,Y(s),Y(sς))ξ(s,U(s),U(sς))]ds+1Γ(q)0t(ts)q1[ζ(s,Y(s),Y(sς))ζ(s,U(s),U(sς))]dw(s)+k=1n[Ik(Y(tk))Ik(U(tk))].

Similarly to the proof of Existence, due to hypotheses(1)-(2), by applying Jensen’s inequality, Doob’s martingale inequality and Cauchy-Schwarz inequality, we derive

E(sup0ut|Y(u)U(u)|2)3Γ2(q)E(sup0ut|0u(us)q1[ξ(s,Y(s),Y(sς))ξ(s,U(s),U(sς))]ds|2)+3Γ2(q)E(sup0ut|0u(us)q1[ζ(s,Y(s),Y(sς))ζ(s,U(s),U(sς))]dw(s)|2)+3E(sup0ut|k=1n[Ik(Y(tk))Ik(U(tk))]|2)3TΓ2(q)E(0t(ts)2q2|ξ(s,Y(s),Y(sς))ξ(s,U(s),U(sς))|2ds)+12Γ2(q)E(0t(ts)2q2|ζ(s,Y(s),Y(sς))ζ(s,U(s),U(sς))|2ds)+3nk=1ndk2E(sup0ut|Y(tk)U(tk))|2),

then

E(sup0ut|Y(u)U(u)|2)(3T+12)L1Γ2(q)0t(ts)2q2E(|Y(s)U(s)|2+|Y(sς)U(sς)|2)ds+3nk=1ndk2E(sup0ut|Y(uk)U(uk))|2)2(3T+12)L1Γ2(q)0t(ts)2q2E(sup0ηs|Y(η)U(η)|2)ds+3mk=1mdk2E(sup0ut|Y(uk)U(uk))|2).

From the Lemma 2.6, we conclude

E(sup0ut|Y(u)U(u)|2)=0,

which indicates that Y(t) = U(t) for all t ∈ (tn, tn + 1]. Hence the solution of system (1.1) is almost surely unique on [−ς, T]. The proof of the whole theorem is completed. □

4. Finite-time stability

In this part, we present two theorems to show the finite-time stability of system (1.1).

Theorem 4.1.

Suppose that hypotheses (1)-(2) hold, and there exist two positive numbers δ, σ satisfying δ < σ, k=1mdk2<1/(4m) and E(|Φ0|2)δ, then system (1.1) is finite-time stable on [−ς, T], provided that

4.1 n=1i=0n(ni)C4ni[C5Γ(2q1)]iΓ(i(2q1)+ni)0T(Ts){i(2q1)(i+1n)}ds<σC31,

where C3, C4, C5 are defined as follows:

C3=4E(|Φ0|2)+4T2qL2(2q1)Γ2(q)+16T2q1L2(2q1)Γ2(q)14mk=1mdk2,C4=8T2q1L2(14mk=1mdk2)(2q1)Γ2(q),C5=32L2(14mk=1mdk2)Γ2(q).

Proof.

For all t ∈ (tn, tn + 1], 0 ≤ nm, by applying Jensen’s inequality, hypotheses (1)–(2), Cauchy-Schwarz inequality and Doob’s martingale inequality, we derive

E(sup0ut|Y(u)|2)4E(|Φ0|2)+4E(sup0ut|k=1nIk(Y(uk))|2)+4Γ2(q)E(sup0ut|0u(us)q1ξ(s,Y(s),Y(sς))ds|2)+4Γ2(q)E(sup0ut|0u(us)q1ζ(s,Y(s),Y(sς))dw(s)|2):=I1+I2+I3+I4,

where

I1=4E(|Φ0|2)4δ,I24mk=1mdk2E(sup0ut|Y(u)|2),I34Γ2(q)E(0t(ts)2q2ds0t|ξ(s,Y(s),Y(sς))|2ds)4T2q1L2(2q1)Γ2(q)E(0t(1+|Y(s)|2+|Y(sς)|2)ds)4T2qL2(2q1)Γ2(q)+4T2q1L2(2q1)Γ2(q)E(0t(|Y(s)|2+|Y(sς)|2)ds),

and

I416Γ2(q)E(0t(ts)2q2|ζ(s,Y(s),Y(sς))|2ds)16L2Γ2(q)E(0t(ts)2q2(1+|Y(s)|2+|Y(sς)|2)ds)16T2q1L2(2q1)Γ2(q)+16L2Γ2(q)E(0t(ts)2q2(|Y(s)|2+|Y(sς)|2)ds).

Therefore, we get

E(sup0ut|Y(u)|2)4E(|Φ0|2)+4mk=1mdk2E(sup0ut|Y(u)|2)+4T2qL2(2q1)Γ2(q)+4T2q1L2(2q1)Γ2(q)E(0t(|Y(s)|2+|Y(sς)|2)ds)+16T2q1L2(2q1)Γ2(q)+16L2Γ2(q)E(0t(ts)2q2(|Y(s)|2+|Y(sς)|2)ds).

Letting h(t)=supςθt|Y(θ)|2, we derive

h(s)=supςθs|Y(θ)|2|Y(s)|2,

and

|Y(sς)|2supςθsς|Y(θ)|2supςθs|Y(θ)|2=h(s),

then

|Y(s)|2h(s),|Y(sς)|2h(s).

Therefore,

E(sup0ut|Y(u)|2)4E(|Φ0|2)+4T2qL2(2q1)Γ2(q)+16T2q1L2(2q1)Γ2(q)14mk=1mdk2+8T2q1L2(14mk=1mdk2)(2q1)Γ2(q)0tE(h(s))ds+32L2(14mk=1mdk2)Γ2(q)0t(ts)2q2E(h(s))ds.

For all θ ∈ [0, t], we have the following:

E(sup0uθ|Y(u)|2)C3+C40θE(h(s))ds+C50θ(θs)2q2E(h(s))dsC3+C40tE(h(s))ds+C50t(ts)2q2E(h(s))ds.

Thus,

E(h(t))=E(supθ[ς,t]|Y(θ)|2)max{E(supθ[ς,0]|Y(θ)|2),E(supθ[0,t]|Y(θ)|2)}max{δ,C3+C40tE(h(s))ds+C50t(ts)2q2E(h(s))ds}.

According to the Lemmas 2.7 and (4.1), we obtain

E(h(t))C3+n=1i=0n(ni)C4niC5i[Γ(2q1)]iΓ(i(2q1)+ni)0t(ts){i(2q1)(i+1n)}C3ds<σ.

Therefore

E(sup0uθ|Y(u)|2)E(supθ[ς,t]|Y(θ)|2)=E(h(t))<σ.

By the Definition 6, the system (1.1) is finite-time stable. This complete the proof. □

Theorem 4.2.

Let Y(t) be the solution of system (1.1), and assume that hypotheses (1)–(2) hold, q ∈ (3/4, 1), and 4mk=1mdk2+8T2qL2(2q1)Γ2(q)+323T2q1L2(4q3)1/2Γ2(q)<1, then the following hold:

M=E(supςtT|Y(t)|2)max{Φ,4E(|Φ0|2)+4T2qL2(2q1)Γ2(q)+16T2q1L2(2q1)Γ2(q)+163T2q1L2(4q3)1/2Γ2(q)1(4mk=1mdk2+8T2qL2(2q1)Γ2(q)+323T2q1L2(4q3)1/2Γ2(q))}.

Proof.

For all t ∈ (tn, tn + 1], 0 ≤ nm, from Theorem 4.1, we know that

E(sup0ut|Y(u)|2)4E(|Φ0|2)+4mk=1mdk2E(sup0ut|Y(u)|2)+4T2qL2(2q1)Γ2(q)+4T2q1L2(2q1)Γ2(q)E(0t(|Y(s)|2+|Y(sς)|2)ds)+16T2q1L2(2q1)Γ2(q)+16L2Γ2(q)E(0t(ts)2q2(|Y(s)|2+|Y(sς)|2)ds),

then we have

E(sup0ut|Y(u)|2)4E(|Φ0|2)+4mk=1mdk2M+4T2qL2(2q1)Γ2(q)+8T2qL2(2q1)Γ2(q)M+16T2q1L2(2q1)Γ2(q)+16L2Γ2(q)E(0t(ts)2q2(|Y(s)|2+|Y(sς)|2)ds).

The last term of the above inequality has the following result, using Hölder inequality and Jensen’s inequality

16L2Γ2(q)E(0t(ts)2q2(|Y(s)|2+|Y(sς)|2)ds) 16L2Γ2(q)(0t(ts)4q4)1/2E(0t(1+|Y(s)|2+|Y(sς)|2)2ds)1/2 16T2q3/2L2(4q3)1/2Γ2(q)E(0t(3+3|Y(s)|4+3|Y(sς)|4)ds)1/2,

by virtue of the mean value theorem of integrals, and the basic inequality a+b+c|a|+|b|+|c|, we know that there exists a constant η ∈ [0, t] such that

16L2Γ2(q)E(0t(ts)2q2(|Y(s)|2+|Y(sς)|2)ds) 16T2q3/2L2(4q3)1/2Γ2(q)E(T(3+3|Y(η)|4+3|Y(ης)|4))1/2 163T2q1L2(4q3)1/2Γ2(q)E(1+|Y(η)|2+|Y(ης)|2) 163T2q1L2(4q3)1/2Γ2(q)+323T2q1L2(4q3)1/2Γ2(q)M.

Therefore

4.2 E(sup0ut|Y(u)|2)4E(|Φ0|2)+4mk=1mdk2M+4T2qL2(2q1)Γ2(q)+8T2qL2(2q1)Γ2(q)M+16T2q1L2(2q1)Γ2(q)+163T2q1L2(4q3)1/2Γ2(q)+323T2q1L2(4q3)1/2Γ2(q)M.

Let M1 = ||Φ||, M2=E(sup0tT|Y(t)|2), there are two cases:

  • If M1M2, then E(supςtT|Y(t)|2)=M1, which gives the theorem 4.2.

  • If M1 < M2, then from (4.2)

    M24E(|Φ0|2)+4T2qL2(2q1)Γ2(q)+16T2q1L2(2q1)Γ2(q)+163T2q1L2(4q3)1/2Γ2(q)1(4mk=1mdk2+8T2qL2(2q1)Γ2(q)+323T2q1L2(4q3)1/2Γ2(q)).

The proof is completed. □

5. Example

In this part, we will give the following examples to demonstrate previous main theoretical results.

Example 1.

Consider the following IFSDEs with time delays:

5.1 {cD0+45Y(t)=0.1Y(t)1+Y2(t)0.1Y(tς)+0.1Y(t)dw(t)dt,t[0,T],ttk,Y(tk+)=Y(tk)+0.05,k=1,2,,m,Y(t)=0,t[ς,0],

where ξ(t,Y(t),Y(tς))=0.1Y(t)1+Y2(t)0.1Y(tς) and ζ(t,Y(t),Y(tς))=0.1Y(t). Letting L1 = L2 = 0.02, T = 2, ς = 1, m = 16, δ = 1, σ = 22. We can derive that the coefficients satisfy the conditions of Theorem 3.1. Therefore, system (5.1) has a unique solution. By Mathematica software, we get C3 = 1.0649, C4 = 0.3549 and C5 = 0.5902. For 1 ≤ x < +∞, let

F(x)=n=1xi=0n(ni)C4ni[C5Γ(2q1)]iΓ(i(2q1)+ni)0T(Ts){i(2q1)(i+1n)}ds.

We can see from Figure 1

n=1i=0n(ni)C4ni[C5Γ(2q1)]iΓ(i(2q1)+ni)0T(Ts){i(2q1)(i+1n)}ds18.8.
Figure 1. The graph of F(x)
Figure 1.

The graph of F(x)

Meanwhile, we have

σC3119.6592>18.8.

Therefore, addressed equations satisfy the conditions of Theorem 4.1. Then system (5.1) is finite-time stable on [−1, 2].

Example 2.

Consider the following IFSDEs with time delays:

5.2 {cD0+34Y(t)=0.1Y(t)0.1Y(tς)+[0.1sinY(t)0.1cosY(tς)]dw(t)dt, t[0,T], ttk,Y(tk+)=Y(tk)+0.05,k=1,2,,m,Y(t)=0.1cos(t),t[ς,0],

where ξ(t,Y(t),Y(tς))=0.1Y(t)0.1Y(tς) and ζ(t,Y(t),Y(tς))=0.1sinY(t)0.1cosY(tς). Letting L1 = L2 = 0.02, we can easily verify that the equations satisfy hypotheses (1)-(2), obviously its satisfy the conditions of Theorem 3.1. Therefore, above system has a unique solution. In addition, letting T = (5π)/8, m = 9, ς = π/8, δ = 1, σ = 26. We get C3 = 1.4175, C4 = 0.3280, C5 = 0.4684 and from following Figure 2, we obtain

n=1i=0n(ni)C4ni[C5Γ(2q1)]iΓ(i(2q1)+ni)0T(Ts){i(2q1)(i+1n)}ds16.8.
Figure 2. The graph of F(x)
Figure 2.

The graph of F(x)

Also, it can be readily deduced that if the conditions of Theorem 4.1 are satisfied in above equations, then from Theorem 4.1 follows that equations (5.2) are finite-time stable on [−π/8, (5π)/8].

6. Conclusions

In this article, we consider IFSDEs with time delays. The system studied in this paper considers the influence of impulses, which is more general than the related references mentioned in the literatures [15; 37; 42; 17; 48]. By applying Picard-Lindelöf method of successive approximation scheme, generalized Grönwall-Bellman inequality, Jensen’s inequality, Doob’s martingale inequality, Itô isometry, Cauchy-Schwarz inequality, and Definition 6, the finite-time stability and existence results for the addressed equations are derived. As an application, we give two examples at the end of this paper to expound proposed criteria. Therefore, those results will enrich the relevant literatures in dealing with IFSDEs. In future work, Laplace transform and its inverse will be used to solve fractional stochastic differential equations with impulses.


(Communicated by Michal Fečkan)


Funding statement: This work was supported by the Natural Science Special Research Fund Project of Guizhou University, China (202002).

Acknowledgement

The authors want to thank the reviewer for spending their precious time to read this manuscript and its insightful comments that helped improve the presentation of this article.

REFERENCES

[1] ABOUAGWA, M.—CHENG, F.—LI, J.: Impulsive stochastic fractional differential equations driven by fractional Brownian motion, Adv. Difference Equ. 2020(1) (2020), Art. No. 57.10.1186/s13662-020-2533-2Search in Google Scholar

[2] ABOUAGWA, M.—LI, J.: Approximation properties for solutions to Itô Doob stochastic fractional differential equations with non-Lipschitz coefficients, Stoch. Dyn. 19(4) (2019), Art. ID 1950029.10.1142/S0219493719500291Search in Google Scholar

[3] AHMAD, M.—ZADA, A.—AHMAD, J.—MOHAMED, A.: Analysis of Stochastic Weighted Impulsive Neutral ψ-Hilfer Integro-Fractional Differential System with Delay, Math. Probl. Eng. 2022 (2022), Art. ID 1490583.10.1155/2022/1490583Search in Google Scholar

[4] AHMADOVA, A.—MAHMUDOV, N.: Existence and uniqueness results for a class of fractional stochastic neutral differential equations, Chaos Solitons Fractals 139 (2020), Art. ID 110253.10.1016/j.chaos.2020.110253Search in Google Scholar

[5] BAINOV, D.—SIMEONOV, P.: Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical Group London, 1993.Search in Google Scholar

[6] BENCHOHRA, M.—HENDERSON, J.—NTOUYAS, S.: Impulsive Differential Equations and Inclusions, Hindawi, New York, 2006.10.1155/9789775945501Search in Google Scholar

[7] CHADHA, A.—PANDEY, D.: Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Anal. 128 (2015), 149–175.10.1016/j.na.2015.07.018Search in Google Scholar

[8] DU, F.—LU, J.: Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities, Appl. Math. Comput. 375 (2020), Art. ID 125079.10.1016/j.amc.2020.125079Search in Google Scholar

[9] DU, F.—LU, J.: New criterion for finite-time stability of fractional delay systems, Appl. Math. Lett. 104 (2020), Art. ID 106248.10.1016/j.aml.2020.106248Search in Google Scholar

[10] DU, F.—LU, J.: New criteria on finite-time stability of fractional-order hopfield neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst. 32(9) (2021), 3858–3866.10.1109/TNNLS.2020.3016038Search in Google Scholar PubMed

[11] DU, F.—LU, J.: New criterion for finite-time synchronization of fractional order memristor-based neural networks with time delay, Appl. Math. Comput. 389 (2021), Art. ID 125616.10.1016/j.amc.2020.125616Search in Google Scholar

[12] FEČKAN, M.—Wang, J.: Periodic impulsive fractional differential equations, Adv. Nonlinear Anal. 8(1) (2019), 482–496.10.1515/anona-2017-0015Search in Google Scholar

[13] FEČKAN, M.—Zhou, Y.—Wang, J.: On the concept and existence of solution for impulsive fractional differential equations, tions. Commun. Nonlinear Sci. Numer. Simul. 17(7) (2012), 3050–3060.10.1016/j.cnsns.2011.11.017Search in Google Scholar

[14] FERHAT, M.—BLOUHI, T.: Existence and uniquenes results for systems of impulsive functional stochastic differential equations driven by fractional Brownian motion with multiple delay, Topol. Methods Nonlinear Anal. 52(2) (2018), 449–476.10.12775/TMNA.2018.009Search in Google Scholar

[15] GUO, T.—JIANG, W.: Impulsive fractional functional differential equations, Comput. Math. Appl. 64(10) (2012), 3414–3424.10.1016/j.camwa.2011.12.054Search in Google Scholar

[16] HADDAD, W.—CHELLABOINA, V.—NERSESOV, S.: Impulsive and Hybrid Dynamical Systems, Princeton University Press, 2006.10.1515/9781400865246Search in Google Scholar

[17] HEI, X.—WU, R.: Finite-time stability of impulsive fractional-order systems with time-delay, Appl. Math. Model. 40(7–8) (2016), 4285–4290.10.1016/j.apm.2015.11.012Search in Google Scholar

[18] KALAMANI, P.—BALEANU, D.—SELVARASU, S.—ARJUNAN, M.: On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Difference Equ. 2016 (2016), Art. No. 163.10.1186/s13662-016-0885-4Search in Google Scholar

[19] KILBAS, A.—SRIVASTAVA, H.—TRUJILLO, J.: Theory and Applications of Fractional Differential Equations, Elsevier: New York, 2006.Search in Google Scholar

[20] KONG, F.—ZHU, Q.: New fixed-time synchronization control of discontinuous inertial neural networks via indefinite Lyapunov-Krasovskii functional method, Internat. J. Robust Nonlinear Control 31(2) (2021), 471–495.10.1002/rnc.5297Search in Google Scholar

[21] KONG, F.—ZHU, Q.—SAKTHIVEL, R.—MOHAMMADZADEH, A.: Fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks with parameter uncertainties, Neurocomputing 422 (2021), 295–313.10.1016/j.neucom.2020.09.014Search in Google Scholar

[22] LAKSHMIKANTHAM, V.—BAINOV, D.—SIMEONOV, P.: Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.10.1142/0906Search in Google Scholar

[23] LI, M.—WANG, J.: Finite time stability of fractional delay differential equations, Appl. Math. Lett. 64 (2017), 170–176.10.1016/j.aml.2016.09.004Search in Google Scholar

[24] LI, M.—WANG, J.: Analysis of nonlinear Hadamard fractional differential equations via properties of Mittag-Leffler functions, J. Appl. Math. Comput. 51(1–2) (2016), 487–508.10.1007/s12190-015-0916-4Search in Google Scholar

[25] LI, M.—WANG, J.: Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput. 324 (2018), 254–265.10.1016/j.amc.2017.11.063Search in Google Scholar

[26] LI, M.—WANG, J.: Finite time stability and relative controllability of Riemann-Liouville fractional delay differential equations, Math. Methods Appl. Sci. 42(18) (2019), 6607–6623.10.1002/mma.5765Search in Google Scholar

[27] LI, Q.—LUO, D.—LUO, Z.—ZHU, Q.: On the novel finite-time stability results for uncertain fractional delay differential equations involving noninstantaneous impulses, Math. Probl. Eng. 2019 (2019), Art. ID 9097135.10.1155/2019/9097135Search in Google Scholar

[28] LIANG, C.—WANG, J.—O’REGAN, D.: Representation of a solution for a fractional linear system with pure delay, Appl. Math. Lett. 77 (2018), 72–78.10.1016/j.aml.2017.09.015Search in Google Scholar

[29] LUO, D.—LUO, Z.: Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Difference Equ. 2019 (2019), Art. No. 155.10.1186/s13662-019-2101-9Search in Google Scholar

[30] LUO, D.—LUO, Z.: Uniqueness and novel finite-time stability of solutions for a class of nonlinear fractional delay difference systems, Discrete Dyn. Nat. Soc. 2018 (2018), Art. ID 8476285.10.1155/2018/8476285Search in Google Scholar

[31] LUO, D.—ZHU, Q.—LUO, Z.: An averaging principle for stochastic fractional differential equations with time-delays, Appl. Math. Lett. 105 (2020), Art. ID 106290.10.1016/j.aml.2020.106290Search in Google Scholar

[32] LUO, Z.—WANG. J.: Finite time stability analysis of systems based on delayed exponential matrix, J. Appl. Math. Comput. 55(1–2) (2017), 335–351.10.1007/s12190-016-1039-2Search in Google Scholar

[33] LUO, Z.—WEI, W.—WANG, J.: Finite time stability of semilinear delay differential equations, Nonlinear Dyn. 89(1) (2017), 713–722.10.1007/s11071-017-3481-6Search in Google Scholar

[34] MAO, X.: Stochastic Differential Equations and Applications, Woodhead Publishing Limited published, Cambridge, 2010.10.1533/9780857099402.47Search in Google Scholar

[35] MILLER, K.—ROSS, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Search in Google Scholar

[36] MITROPOLSKIY, Y.—IOVANE, G.—BORYSENKO, S.: About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications, Nonlinear Anal. 66(10) (2007), 2140–2165.10.1016/j.na.2006.03.006Search in Google Scholar

[37] MOGHADDAM, B.—ZHANG, L.—LOPES, A.—TENREIRO MACHADO J.—MOSTAGHIM, Z.: Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, Stochastics 92(3) (2019), 379–396.10.1080/17442508.2019.1625903Search in Google Scholar

[38] OLDHAM, K.—SPANIER, J.: The Fractional Calculus, San Diego Academic Press, New York, 1974.Search in Google Scholar

[39] PRATO, G.—ZABCZYK, J.: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.Search in Google Scholar

[40] SARAVANAKUMAR, T.—MUOI, N.—ZHU, Q.: Finite-time sampled-data control of switched stochastic model with non-deterministic actuator faults and saturation nonlinearity, J. Franklin Inst. 357(18) (2020), 13637–13665.10.1016/j.jfranklin.2020.10.018Search in Google Scholar

[41] SHAH, K.—ZADA, A.: Controllability and stability analysis of an oscillating system with two delays, Math Meth Appl Sci. 44(18) (2021), 14733–14765.10.1002/mma.7739Search in Google Scholar

[42] UMAMAHESWARI, P.—BALACHANDRAN, K.—ANNAPOORANI, N.: Existence and stability results for Caputo fractional stochastic differential equations with Lévy noise, Filomat 34(5) (2020), 1739–1751.10.2298/FIL2005739USearch in Google Scholar

[43] WANG, J.—FEČKAN, M.—Zhou, Y.: Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (2012), 3389–3405.10.1016/j.camwa.2012.02.021Search in Google Scholar

[44] WANG, J.—FEČKAN, M.—Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011), 345–361.10.4310/DPDE.2011.v8.n4.a3Search in Google Scholar

[45] WANG, J.—FEČKAN, M.—Zhou, Y.: Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. Spec. Top. 222 (2013), 1857–1874.10.1140/epjst/e2013-01969-9Search in Google Scholar

[46] WANG, J.—FEČKAN, M.—Zhou, Y.: A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal. 19(4) (2016), 806–831.10.1515/fca-2016-0044Search in Google Scholar

[47] WANG, J.—LUO, Z.: Finite time stability of semilinear multi-delay differential systems, Trans. Inst. Meas. Control. 40(9) (2017), 2948–2959.10.1177/0142331217711749Search in Google Scholar

[48] WANG, X.—LUO, D.—LUO, Z.—ZADA, A.: Ulam-Hyers stability of Caputo-type fractional stochastic differential equations with time delays, Math. Probl. Eng. 2021 (2021), Art. ID 5599206.10.1155/2021/5599206Search in Google Scholar

[49] WU, Q. A new type of the Gronwall-Bellman inequality and its application to fractional stochastic differential equations, Cogent Math. Stat. 4(1) (2017), Art. ID 1279781.10.1080/23311835.2017.1279781Search in Google Scholar

[50] YOU, Z.—WANG, J.—ZHOU, Y.—FEČKAN, M.: Representation of solutions and finite time stability for delay differential systems with impulsive effects, Int. J. Nonlinear Sci. Numer. Simul. 20(2) (2019), 205–221.10.1515/ijnsns-2018-0137Search in Google Scholar

[51] ZADA, A.— ALI, W.—PARK, C.: Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Gronwall-Bellman-Bihari’s type, Appl. Math. Comput. 305 (2019), 60–65.10.1016/j.amc.2019.01.014Search in Google Scholar

[52] ZADA, A.—PERVAIZ, B.—SUBRAMANIAN, M.—POPA, I.: Finite time stability for nonsingular impulsive first order delay differential systems, Appl. Math. Comput. 421 (2022), Art. ID 126943.10.1016/j.amc.2022.126943Search in Google Scholar

[53] ZHANG, Y.—WANG, J.: Existence and finite-time stability results for impulsive fractional differential equations with maxima, J. Appl. Math. Comput. 51(1–2) (2016), 67–79.10.1007/s12190-015-0891-9Search in Google Scholar

Received: 2022-02-17
Accepted: 2022-04-01
Published Online: 2023-03-31
Published in Print: 2023-04-01

© 2023 Mathematical Institute Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/ms-2023-0030/html
Scroll to top button